Question on functions (absolute fns.)

rock.freak667

Homework Helper
6,230
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1. The problem statement, all variables and given/known data

[tex]f:\rightarrow x+1,g:x\rightarrow |x|[/tex]

Solve the equation gf(x)=fg(x)
2. Relevant equations



3. The attempt at a solution


gf(x)=|x+1|
and fg(x)=|x|+1

so I drew the graphs of y=gf(x) and y=fg(x) on the same axes.
For x<0 the graphs do not intersect as the two lines are parallel (having the same gradient) and hence there is no solution for x<0.

BUT, for x>0, the two lines are the same...so that means there are an infinite number of solutions for x>0. Does that mean I write the answer as {x:x>0} ?
 

HallsofIvy

Science Advisor
41,618
821
Almost. If x= 0, |0+1|= 1 so x= 0 also satisfies the equation. In particular, if x> 0, |x|+ 1= x+ 1 and since x>0>-1, |x+1|= x+ 1 so the equation |x|+1= |x+1| is the same as x+1= x+1. That's satisfied for all x so |x+ 1|= |x|+ 1 is satisfied for all [itex]x\ge 0[/itex] (not "x> 1").
 

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